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Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions and quasi-concave functions. Under weaker partial orders on non-negative functions, monotone envelopes are re-examined and the level function is recognized as a monotone envelope in two ways. Using the level function, monotonicity can be transferred from the kernel to the weight in inequalities restricted to a cone of monotone functions.
@article{Sinnamon2003, abstract = {Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions and quasi-concave functions. Under weaker partial orders on non-negative functions, monotone envelopes are re-examined and the level function is recognized as a monotone envelope in two ways. Using the level function, monotonicity can be transferred from the kernel to the weight in inequalities restricted to a cone of monotone functions.}, author = {Sinnamon, Gord}, journal = {Collectanea Mathematica}, keywords = {Desigualdades; Espacios de funciones medibles; Monotonicidad; Funciones de peso; quasi-concave functions; monotonicity; weight; inequalities}, language = {eng}, number = {2}, pages = {181-216}, title = {Transferring monotonicity in weighted norm inequalities.}, url = {http://eudml.org/doc/43083}, volume = {54}, year = {2003}, }
TY - JOUR AU - Sinnamon, Gord TI - Transferring monotonicity in weighted norm inequalities. JO - Collectanea Mathematica PY - 2003 VL - 54 IS - 2 SP - 181 EP - 216 AB - Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions and quasi-concave functions. Under weaker partial orders on non-negative functions, monotone envelopes are re-examined and the level function is recognized as a monotone envelope in two ways. Using the level function, monotonicity can be transferred from the kernel to the weight in inequalities restricted to a cone of monotone functions. LA - eng KW - Desigualdades; Espacios de funciones medibles; Monotonicidad; Funciones de peso; quasi-concave functions; monotonicity; weight; inequalities UR - http://eudml.org/doc/43083 ER -